The thing that is confusing you is not true. Both sums have n-1 terms.

Did you try seeing what happens with specific n? If n= 5, the sum from 1 to n-1= 4 is [itex]a_1+ a_2+ a_3+ a_4[/itex]. That has 4= n-1 terms, not n-2.

The sum from 1 to infinity would be [itex]1+ a_1+ a_2+ a_3+ a_4+ a_ 5+ a_6+ a_7+ \cdot\cdot\cdot+ [/itex] while the sum from n to inifinity is [itex]a_5+ a_6+ a_7+ \cdot\cdot\cdot[/itex]. Subtracting the second from the first leaves [itex]a_1+ a_2+ a_3+ a_4[/itex] as claimed.